Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $z = \dfrac{10}{5(5a - 7)} \div \dfrac{-8}{15a - 21} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{10}{5(5a - 7)} \times \dfrac{15a - 21}{-8} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ 10 \times (15a - 21) } { 5(5a - 7) \times -8 } $ $ z = \dfrac {10 \times 3(5a - 7)} {-8 \times 5(5a - 7)} $ $ z = \dfrac{30(5a - 7)}{-40(5a - 7)} $ We can cancel the $5a - 7$ so long as $5a - 7 \neq 0$ Therefore $a \neq \dfrac{7}{5}$ $z = \dfrac{30 \cancel{(5a - 7})}{-40 \cancel{(5a - 7)}} = -\dfrac{30}{40} = -\dfrac{3}{4} $